The meson-baryon molecular components for the $N^{*}$ and ${\mathrm{\Delta }}^{*}$ resonances are investigated in terms of the compositeness, which is defined as the norm of the two-body wave function from the meson-baryon scattering amplitudes. The scattering amplitudes are constructed in a $\pi N\text{−}\eta N\text{−}\sigma N\text{−}\rho N\text{−}\pi \mathrm{\Delta }$ coupled-channels problem in a meson exchange model together with several bare $N^{*}$ and ${\mathrm{\Delta }}^{*}$ states, and parameters are fitted so as to reproduce the on-shell $\pi N$ partial wave amplitudes up to the center-of-mass energy $1.9\mathrm{GeV}$ with the orbital angular momentum $L\le 2$. As a result, the Roper resonance $N\left(1440\right)$ is found to be dominated by the $\pi N$ and $\sigma N$ molecular components while the bare-state contribution is small. The squared wave functions in coordinate space imply that both in the $\pi N$ and $\sigma N$ channels the separation between the meson and baryon is about more than $1\mathrm{fm}$ for the $N\left(1440\right)$ resonance. On the other hand, dominant meson-baryon molecular components are not observed in any other $N^{*}$ and ${\mathrm{\Delta }}^{*}$ resonances in the present model, although they have some fractions of the meson-baryon clouds.

中文翻译：

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散射振幅的二体波函数和复合性。二、在物理N*和Δ*共振中的应用

介子重子分子组成 $N^{*}$ 和 ${\mathrm{\Delta }}^{*}$根据复合性研究共振，复合性被定义为介子重子散射振幅的二体波函数的范数。散射振幅被构造为$\pi N\text{-}\eta N\text{-}\sigma N\text{-}\rho N\text{-}\pi \mathrm{\Delta }$ 介子交换模型中的耦合通道问题与几个裸 $N^{*}$ 和 ${\mathrm{\Delta }}^{*}$ 状态，并拟合参数以重现壳上 $\pi N$ 部分波振幅达到质心能量 $1.9\mathrm{电子伏特}$ 与轨道角动量 $升\le 2$. 因此，罗珀共振$N\left(1440\right)$ 被发现是由 $\pi N$ 和 $\sigma N$分子成分，而裸态贡献很小。坐标空间中的平方波函数意味着$\pi N$ 和 $\sigma N$ 介子和重子之间的间隔大约大于 $1\mathrm{调频}$ 为了 $N\left(1440\right)$谐振。另一方面，在任何其他物质中都没有观察到占主导地位的介子重子分子成分。$N^{*}$ 和 ${\mathrm{\Delta }}^{*}$ 本模型中的共振，尽管它们具有介子重子云的一些部分。